Applying Ito's Formula to F(Z_{t}, S_{t}, B_{t})

[InvisibleBlue]

Hi,

I'm trying to answer this question on why we can apply Ito's formula to the function $$F(Z_{t}, S_{t}, B_{t})$$ which is a $$C^{1,1,2}$$ function where:

$$B_{t}$$ is standard brownian motion

$$S_{t}=max_{0\leq s\leq t}B_{s}$$
and
$$Z_{t}= \int_0^t B_{s} ds$$

I think I basically have to show that $$S_{t}$$ and $$Z_{t}$$ are continuous and of bounded variation. But can't quite see how to show either, specially in the case of $$S_{t}$$ since we know that the standard brownian motion is not of bounded variation.

any ideas?

 

[Valkarie]

Thanks!To show that S_{t} and Z_{t} are continuous and of bounded variation, you can use the fact that Brownian motion is a continuous semimartingale. This means that for any t > 0, B_{t} is a continuous process, and thus so is S_{t}. In addition, the integral form of Z_{t} is itself a continuous function, so Z_{t} is also continuous. For bounded variation, we can use the fact that Brownian motion is a semimartingale. This means that the quadratic variation of B_{t} is equal to t. Thus, since S_{t} is just a maximum of t values of B_{t}, it follows that the quadratic variation of S_{t} is also equal to t. Similarly, since the integral form of Z_{t} is a function of t values of B_{t}, the quadratic variation of Z_{t} is also equal to t. Thus, both S_{t} and Z_{t} have bounded variation.